Integrand size = 39, antiderivative size = 310 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}+\frac {1}{6} \text {RootSum}\left [a^9-2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 b^2 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}-\text {$\#$1}^4}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(1777\) vs. \(2(310)=620\).
Time = 1.50 (sec) , antiderivative size = 1777, normalized size of antiderivative = 5.73, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1607, 2081, 6857, 129, 494, 245, 384} \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{b}-\sqrt [3]{-2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \]
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Rule 129
Rule 245
Rule 384
Rule 494
Rule 1607
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (b+a x^2\right )}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx \\ & = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x} \left (b+a x^2\right )}{\sqrt [3]{b^2+a^3 x} \left (b+2 a x^3\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (-\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {\left (-\frac {\sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (-\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {\left (-\frac {(-1)^{2/3} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (\left (\sqrt [3]{-2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (\left (\sqrt [3]{-2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = -\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{-\frac {1}{2}} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{-\frac {1}{2}} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 10.42 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.86 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {3 \sqrt [3]{a} x \sqrt [3]{1+\frac {a^3 x}{b^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {a^3 x}{b^2}\right )-x \left (\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}+\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {a^3 x-\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{a} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
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Time = 0.54 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {-6 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}+2 a \,b^{5}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right ) \left (\textit {\_R}^{3}-a^{3}-2 b^{2}\right )}{\textit {\_R} \left (\textit {\_R}^{3}-a^{3}\right )}\right ) a -6 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+3 \ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{12 a}\) | \(212\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.39 (sec) , antiderivative size = 66610, normalized size of antiderivative = 214.87 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 23.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.10 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {x \left (a x^{2} + b\right )}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{3} + b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.13 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{3} + b x}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{3} + b\right )}} \,d x } \]
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Timed out. \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Timed out} \]
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Not integrable
Time = 7.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.13 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a\,x^3+b\,x}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a\,x^3+b\right )} \,d x \]
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